Anisotropic Hyperelastic Strain Energy Function for Carbon Fiber Woven Fabrics.

The present paper introduces an innovative strain energy function (SEF) for incompressible anisotropic fiber-reinforced materials. This SEF is specifically designed to understand the mechanical behavior of carbon fiber-woven fabric. The considered model combines polyconvex invariants forming an integrity basisin polynomial form, which is inspired by the application of Noether's theorem. A single solution can be obtained during the identification because of the relationship between the SEF we have constructed and the material parameters, which are linearly dependent. The six material parameters were precisely determined through a comparison between the closed-form solutions from our model and the corresponding tensile experimental data with different stretching ratios, with determination coefficients consistently reaching a remarkable value of 0.99. When considering only uniaxial tensile tests, our model can be simplified from a quadratic polynomial to a linear polynomial, thereby reducing the number of material parameters required from six to four, while the fidelity of the model's predictive accuracy remains unaltered. The comparison between the results of numerical calculations and experiments proves the efficiency and accuracy of the method.

Energetically stable curve fitting to hyperelastic models based on uniaxial and biaxial tensile tests.

Hyperelastic constitutive laws in biomechanics are used to model soft tissues, and material model parameters are often determined by performing curve fitting on data from uniaxial or biaxial tensile tests. The strain energy function of the applied constitutive law must to be energetically stable; however, this condition is not inherently provided by most currently available models. This study provides a procedure to determine stable strain energy functions in a biaxial strain space based on either uniaxial or biaxial tensile tests. Instead of conservative, strain-independent conditions, a stability region is defined in the strain space based on the sample's tensile tests, thus allowing optimisation within a wider parameter space, resulting in better approximations. An extension of the Levenberg-Marquardt algorithm incorporating user-defined stability constraints is proposed, and the constrained optimisation algorithm is applied to isotropic and anisotropic models. The uniqueness of solutions of the Fung model is also discussed. The material model parameters of stable solutions for soft tissue measurements from various literature sources are determined to demonstrate the proposed procedure. Applying appropriate constraints in the optimisation algorithm resulted in stable and physically permissible constrained solutions for the strain energy function, in contrast to the results of most unconstrained optimisation cases.

Some Mechanical Constraints to the Biomimicry with Peripheral Nerves.

Novel high technology devices built to restore impaired peripheral nerves should be biomimetic in both their structure and in the biomolecular environment created around regenerating axons. Nevertheless, the structural biomimicry with peripheral nerves should follow some basic constraints due to their complex mechanical behaviour. However, it is not currently clear how these constraints could be defined. As a consequence, in this work, an explicit, deterministic, and physical-based framework was proposed to describe some mechanical constraints needed to mimic the peripheral nerve behaviour in extension. More specifically, a novel framework was proposed to investigate whether the similarity of the stress/strain curve was enough to replicate the natural nerve behaviour. An original series of computational optimizing procedures was then introduced to further investigate the role of the tangent modulus and of the rate of change of the tangent modulus with strain in better defining the structural biomimicry with peripheral nerves.

Biomechanical constitutive modeling of the gastrointestinal tissues: a systematic review.

The gastrointestinal (GI) tract is a continuous channel through the body that consists of the esophagus, the stomach, the small intestine, the large intestine, and the rectum. Its primary functions are to move the intake of food for digestion before storing and ultimately expulsion of feces. The mechanical behavior of GI tissues thus plays a crucial role for GI function in health and disease. The mechanical properties are characterized by a biomechanical constitutive model, which is a mathematical representation of the relation between load and deformation in a tissue. Hence, validated biomechanical constitutive models are essential to characterize and simulate the mechanical behavior of the GI tract. Here, a systematic review of these constitutive models is provided. This review is limited to studies where a model of the strain energy function is proposed to characterize the stress-strain relation of a GI tissue. Several needs are identified for more advanced modeling including: 1) Microstructural models that provide actual structure-function relations; 2) Validation of coupled electro-mechanical models accounting for active muscle contractions; 3) Human data to develop and validate models. The findings from this review provide guidelines for using existing constitutive models as well as perspective and directions for future studies.

A generalized Ogden model for the compressibility of rubber-like solids.

The aim of this paper is to further demonstrate the advantages and effectiveness of the constitutive formulation proposed by Huang (Huang 2014 J. Appl. Mech. 59, 902-908 (doi:10.1115/1.2894059)). In this formulation, any strain-energy function for an incompressible material can be easily generalized to include the effect of material compressibility, in which only a few material parameters and material functions to be fitted with the experimental data are required. To this end, the Ogden model for incompressible rubber-like solids is chosen as the starting point. By means of this formulation, the generalized Ogden strain-energy function, which takes into account material compressibility, can conveniently be constructed so long as its incompressible counterpart is given. The obvious advantage shown in this paper is that only a few material parameters and material functions are needed, i.e. in addition to the material parameters used in the original Ogden model for incompressible solids, only one material function depending on the volume ratio is involved to characterize the effect of compressibility. Both the material parameters in the original Ogden model and the material function suggested in this paper can be determined by fitting the experimental data for uniaxially tensile test and hydrostatic deformation test of rubbers, respectively. The present model considering compressibility is general since it can be applied to predict the stress-strain responses of rubber-like materials and porous rubbers in various loading conditions. With the present formulation, the applicable range of the celebrated Ogden model can be further broadened, which should be of practical importance for accurately describing the mechanical behaviour of rubber-like solids. This article is part of the theme issue 'The Ogden model of rubber mechanics: Fifty years of impact on nonlinear elasticity'.

Multiscale Contrasts Between the Right and Left Ventricle Biomechanics in Healthy Adult Sheep and Translational Implications.

Cardiac biomechanics play a significant role in the progression of structural heart diseases (SHDs). SHDs alter baseline myocardial biomechanics leading to single or bi-ventricular dysfunction. But therapies for left ventricle (LV) failure patients do not always work well for right ventricle (RV) failure patients. This is partly because the basic knowledge of baseline contrasts between the RV and LV biomechanics remains elusive with limited discrepant findings. The aim of the study was to investigate the multiscale contrasts between LV and RV biomechanics in large animal species. We hypothesize that the adult healthy LV and RV have distinct passive anisotropic biomechanical properties. Ex vivo biaxial tests were performed in fresh sheep hearts. Histology and immunohistochemistry were performed to measure tissue collagen. The experimental data were then fitted to a Fung type model and a structurally informed model, separately. We found that the LV was stiffer in the longitudinal (outflow tract) than circumferential direction, whereas the RV showed the opposite anisotropic behavior. The anisotropic parameter K from the Fung type model accurately captured contrasting anisotropic behaviors in the LV and RV. When comparing the elasticity in the same direction, the LV was stiffer than the RV longitudinally and the RV was stiffer than the LV circumferentially, suggesting different filling patterns of these ventricles during diastole. Results from the structurally informed model suggest potentially stiffer collagen fibers in the LV than RV, demanding further investigation. Finally, type III collagen content was correlated with the low-strain elastic moduli in both ventricles. In summary, our findings provide fundamental biomechanical differences between the chambers. These results provide valuable insights for guiding cardiac tissue engineering and regenerative studies to implement chamber-specific matrix mechanics, which is particularly critical for identifying biomechanical mechanisms of diseases or mechanical regulation of therapeutic responses. In addition, our results serve as a benchmark for image-based inverse modeling technologies to non-invasively estimate myocardial properties in the RV and LV.

Effects of material non-symmetry on the mechanical behavior of arterial wall.

Arteries are commonly assumed as symmetric cylindrical tubes with axisymmetric geometry and mechanical properties. However, their wall stress, structure and mechanical properties may become nonsymmetric when subject to torsion or complex mechanical loading. The objective of this study was to explore the nonsymmetric two fiber family constitutive models for arterial walls and examine the impact of this non-symmetry on the deformation and stress in arteries under mechanical loads. Our results demonstrated that nonsymmetric collagen fiber properties and alignment lead to interesting phenomena such as vessel twisting associated with axial stretch or pressurization. There are "magic" nonsymmetric fiber angles at which a vessel would not twist under given pressure and axial stretch. The nonsymmetric fiber properties and alignment (mean angle and dispersion) affects the torque-twist angle relationship as well as the axial stretch and pressurized inflation. These results illustrate the effects of nonsymmetric collagen fiber distribution and suggest that the Holzapfel-Gasser-Ogden models could be generalized to incorporate the nonsymmetric two fiber families for broader applications, especially when there is shear or torsion.

Effects of dispersed fibres in myocardial mechanics, Part I: passive response.

It is widely acknowledged that an imbalanced biomechanical environment can have significant effects on myocardial pathology, leading to adverse remodelling of cardiac function if it persists. Accurate stress prediction essentially depends on the strain energy function which should have competent descriptive and predictive capabilities. Previous studies have focused on myofibre dispersion, but not on fibres along other directions. In this study, we will investigate how fibre dispersion affects myocardial biomechanical behaviours by taking into account both the myofibre dispersion and the sheet fibre dispersion, with a focus on the sheet fibre dispersion. Fibre dispersion is incorporated into a widely-used myocardial strain energy function using the discrete fibre bundle approach. We first study how different dispersion affects the descriptive capability of the strain energy function when fitting to ex vivo experimental data, and then the predictive capability in a human left ventricle during diastole. Our results show that the chosen strain energy function can achieve the best goodness-of-fit to the experimental data by including both fibre dispersion. Furthermore, noticeable differences in stress can be found in the LV model. Our results may suggest that it is necessary to include both dispersion for myofibres and the sheet fibres for the improved descriptive capability to the ex vivo experimental data and potentially more accurate stress prediction in cardiac mechanics.

3D biomechanical properties of the layered esophagus: Fung-type SEF and new constitutive model.

BACKGROUND AND PURPOSE: Most current studies on the passive biomechanical properties of esophageal tissues directly use the exponential strain energy function (SEF) to fit and calculate the constants of the constitutive equation. In the context of the extensive application of exponential SEF, in-depth research on the exponential SEF is still lacking. The purpose of this study is to combine the exponential function with the polynomial SEF to obtain the most suitable constitutive equation to describe the three-dimensional passive behavior of the esophagus. METHODS: fresh pig esophagus with a length of 13 cm in the middle position was selected as esophageal samples. The esophageal sample was separated into muscular layer and mucosal layer with surgical scissors. Stretch-inflation mechanical tests of the intact esophagus, esophageal muscular, and esophageal mucosa were carried out on a triaxial test machine. The external radius, axial force, and internal pressure were recorded simultaneously. The seven-parameter Fung-type SEF and several new SEFs combining polynomials and exponents were used to fit the experimental data curves. RESULTS: The stretch-inflation test data and the morphometric parameters at the zero-stress state of the layered esophagus were obtained. The new SEF with polynomial and exponential combination is more suitable to describe describing the three-dimensional passive biomechanical properties of esophageal tissue. Among them, New-Fung13 SEF is more suitable for describing the passive biomechanical properties of intact esophageal tissue, Sokolis-Fung13 SEF is more suitable for the esophageal muscle layer, and New-Fung10 SEF is more suitable for the esophageal mucosa. The constitutive parameters of the optimal constitutive model for each layer of the esophagus were obtained.

Age-related changes in mechanical properties of human abdominal fascia.

The purpose of this study is to assess and model age-related changes in the mechanical properties of human fascia. The samples were divided into three age groups: group A-up to 60 years (mean age 52.5 ± 6 years), group B-61-80 years (mean age 70.4 ± 5.2 years), and group C-81-90 years (mean age 83.2 ± 2 years). A uniaxial tensile test was applied to fascia specimens cut perpendicular and parallel to fibers. The secant modulus at 5% strain, the maximum stress, and the stretch at maximum stress were calculated from the stress-stretch ratio curves. The results indicated an increase in the secant modulus with the increased age. The trend is clearer in the longitudinal direction. Considering the strain energy function which accounts the isotropic and non-isotropic response of the fascia where isotropic and anisotropic parts are split, we evaluated which material model is the most suitable to present isotropic mechanical behavior of the tissue. The experimental stress-stretch ratio curves were approximated using Mooney-Rivlin, Yeoh, and neo-Hookean strain energy functions and a good match between theoretical and experimental results was obtained. On the basis of objective function values and normalized mean square root error, we recommend using the Yeoh model to describe the isotropic mechanical behavior of human abdominal fascia. Graphical abstract .

Numerical and Experimental Investigation of Oil Palm Shell Reinforced Rubber Composites.

This paper presents a pioneering effort to ascertain the suitability of hyperelastic modelling in simulating the stress-strain response of oil palm shell reinforced rubber (ROPS) composites. ROPS composites with different oil palm shell contents (0%, 5%, 10% and 20% by volume) were cast in the laboratory for the experimental investigation. ROPS specimens with circular, square, hexagon, and octagon shapes (loading surface) were considered to evaluate the accuracy of finite element simulation considering the shape effect of composites. Strain-controlled (compressive) tests with ε ≈ 50% at 0.8 Hz frequency were conducted in the laboratory and the test data obtained was used as input to simulate material coefficients corresponding to the strain energy functions chosen. Five different strain energy functions were selected and utilized for the hyperelastic modelling in this study using finite element approach. The shape effect was then used to ascertain any variation in the simulation outcomes and to discuss the effect of shape on the behaviour of ROPS composites in comparison to existing literature. The numerical predictions using the Yeoh model (error ≤ 2.7% for circular shaped ROPS) were found to perform best in comparison with the experimental results, thus a more stable and suitable hyperelastic model to this end. The Marlow (error ≤ 4.6% for circular shaped ROPS) and Arruda Boyce (error ≤ 4.7% for circular shaped ROPS) models were amongst the next alternatives to perform better. Even with the other shapes considered in this study, Yeoh, followed by the Marlow function, were more appropriate models. The shape effect was then studied with particular emphasis on comparing and assessing them with that observed in the literature. To this end, adopting the Yeoh function in the finite element model is the ideal approach to estimate the stress-strain response of ROPS composites.

Constitutive law of healthy gallbladder walls in passive state with damage effect.

Biomechanical properties of human gallbladder (GB) wall in passive state can be valuable to diagnosis of GB diseases. In the article, an approach for identifying damage effect in GB walls during uniaxial tensile test was proposed and a strain energy function with the damage effect was devised as a constitutive law phenomenologically. Scalar damage variables were introduced respectively into the matrix and two families of fibres to assess the damage degree in GB walls. The parameters in the constitutive law with the damage effect were determined with a custom MATLAB code based on two sets of existing uniaxial tensile test data on human and porcine GB walls in passive state. It turned out that the uniaxial tensile test data for GB walls could not be fitted properly by using the existing strain energy function without the damage effect, but could be done by means of the proposed strain energy function with the damage effect involved. The stresses and Young moduli developed in two families of fibres were more than thousands higher than the stresses and Young's moduli in the matrix. According to the damage variables estimated, the damage effect occurred in two families of fibres only. Once the damage occurs, the value of the strain energy function will decrease. The proposed constitutive laws are meaningful for finite element analysis on human GB walls.

Computational Modeling of Human Left Ventricle to Assess the Effects of Trabeculae Carneae on the Diastolic and Systolic Functions.

Trabeculae carneae are irregular structures that cover the endocardial surfaces of both ventricles and account for a significant portion of human ventricular mass. The role of trabeculae carneae in diastolic and systolic functions of the left ventricle (LV) is not well understood. Thus, the objective of this study was to investigate the functional role of trabeculae carneae in the LV. Finite element (FE) analyses of ventricular functions were conducted for three different models of human LV derived from high-resolution magnetic resonance imaging (MRI). The first model comprised trabeculae carneae and papillary muscles, while the second model had papillary muscles and partial trabeculae carneae, and the third model had a smooth endocardial surface. We customized these patient-specific models with myofiber architecture generated with a rule-based algorithm, diastolic material parameters of Fung strain energy function derived from biaxial tests and adjusted with the empirical Klotz relationship, and myocardial contractility constants optimized for average normal ejection fraction (EF) of the human LV. Results showed that the partial trabeculae cutting model had enlarged end-diastolic volume (EDV), reduced wall stiffness, and even increased end-systolic function, indicating that the absence of trabeculae carneae increased the compliance of the LV during diastole, while maintaining systolic function.

On the use of machine learning techniques for the mechanical characterization of soft biological tissues.

Motivated by the search for new strategies for fitting a material model, a new approach is explored in the present work. The use of numerical and complex algorithms based on machine learning techniques such as support vector machines for regression, bagged decision trees, and artificial neural networks is proposed for solving the parameter identification of constitutive laws for soft biological tissues. First, the mathematical tools were trained with analytical uniaxial data (circumferential and longitudinal directions) as inputs, and their corresponding material parameters of the Gasser, Ogden, and Holzapfel strain energy function as outputs. The train and test errors show great efficiency during the training process in finding correlations between inputs and outputs; besides, the correlation coefficients were very close to 1. Second, the tool was validated with unseen observations of analytical circumferential and longitudinal uniaxial data. The results show an excellent agreement between the prediction of the material parameters of the strain energy function and the analytical curves. Finally, data from real circumferential and longitudinal uniaxial tests on different cardiovascular tissues were fitted; thus, the material model of these tissues was predicted. We found that the method was able to consistently identify model parameters, and we believe that the use of these numerical tools could lead to an improvement in the characterization of soft biological tissues.

Constitutive laws with damage effect for the human great saphenous vein.

Strain energy-based constitutive laws with damage effect were proposed by using existing both uniaxial tensile test and tubular biaxial inflation test data on the human great saphenous vein (GSV) segments. These laws were applied into GSV coronary artery bypass grafts (CABG) by employing a thin-walled vessel model to evaluate their passive biomechanical performance under coronary artery physiological conditions at a fixed axial pre-stretch. At a peak systolic pressure in 100-150 mmHg, a 20-33% GSV diameter dilation was predicted with the law based on tubular biaxial inflation test data and agreed well with 25% dilation in clinical observation in comparison with as small as 2-4% dilation estimated with the law based on uniaxial tensile test data. The constitutive law generated by tubular biaxial inflation test data was mostly suitable for GSV CABG under coronary artery physiological conditions than that based on uniaxial tensile test results. With these laws, the fibre ultimate stretch was extracted from uniaxial tensile test data and the structural sub-failure/damage threshold of 1.0731 was decided for the human GSV. GSV fibres could exhibit damage effect but unlikely undergo a structure failure/break, suggesting a damage factor might exist during CABG arterialization. The damage in GSV tissue might initiate or contribute to early remodelling of CABG after implantation.

A unified approach to model peripheral nerves across different animal species.

Peripheral nerves are extremely complex biological structures. The knowledge of their response to stretch is crucial to better understand physiological and pathological states (e.g., due to overstretch). Since their mechanical response is deterministically related to the nature of the external stimuli, theoretical and computational tools were used to investigate their behaviour. In this work, a Yeoh-like polynomial strain energy function was used to reproduce the response of in vitro porcine nerve. Moreover, this approach was applied to different nervous structures coming from different animal species (rabbit, lobster, Aplysia) and tested for different amount of stretch (up to extreme ones). Starting from this theoretical background, in silico models of both porcine nerves and cerebro-abdominal connective of Aplysia were built to reproduce experimental data (R2 > 0.9). Finally, bi-dimensional in silico models were provided to reduce computational time of more than 90% with respect to the performances of fully three-dimensional models.

Modelling of compressible and orthotropic surgical mesh implants based on optical deformation measurement.

There is a potential mismatch between surgical mesh implants for hernia repair of pelvic floor surgery and the host tissue because soft tissue is incompressible and meshes are compressible. Therefore, mesh and tissue may develop different stiffness over the range of deformation. In addition compressibility is related to a change of porosity of the mesh which may decrease during the deformation. Scar formation and the ingrowth of the mesh can be related to effective porosity which decreases discontinuously in uniaxial loading at a critical stretch when pore areas collapse and therefore the mesh becomes ineffective. Compressibility requires several non standard approaches which can be performed with high accuracy and local resolution by deformation measurement with digital image correlation (DIC). A compressible hyperelastic model is chosen and identified with biaxial deformation measurements. Also effective porosity of deformed meshes can be calculated on the basis of biaxial deformation. The proposed constitutive equation and the developed model of effective porosity are represented in form of principle stretch. Stretch can be measured with magnetic resonance imaging (MRI) visible meshes so that stress and effective porosity can be derived in vivo.

Importance of material parameters and strain energy function on the wall stresses in the left ventricle.

Patient-specific estimates of the stress distribution in the left ventricles (LV) may have important applications for therapy planning, but computing the stress generally requires knowledge of the material behaviour. The passive stress-strain relation of myocardial tissue has been characterized by a number of models, but material parameters (MPs) remain difficult to estimate. The aim of this study is to implement a zero-pressure algorithm to reconstruct numerically the stress distribution in the LV without precise knowledge of MPs. We investigate the sensitivity of the stress distribution to variations in the different sets of constitutive parameters. We show that the sensitivity of the LV stresses to MPs can be marginal for an isotropic constitutive model. However, when using a transversely isotropic exponential strain energy function, the LV stresses become sensitive to MPs, especially to the linear elastic coefficient before the exponential function. This indicates that in-vivo identification efforts should focus mostly on this MP for the development of patient-specific finite-element analysis.

A Pointwise Method for Identifying Biomechanical Heterogeneity of the Human Gallbladder.

Identifying the heterogeneous biomechanical property of human gallbladder (GB) walls from non-invasive measurements can have clinical significance in patient-specific modeling and acalculous biliary pain diagnosis. In this article, a pointwise method was proposed to measure the heterogeneity of ten samples of human GB during refilling. Three different points, two on the equator of GB body 90° apart and one on the apex of GB fundus, were chosen to represent the typical regions of interest. The stretches at these points were estimated from ultrasound images of the GB during the bile emptying phase based on an analytical model. The model was validated against the experimental data of a lamb GB. The material parameters at the different points were determined inversely by making use of a structure-based anisotropic constitutive model. This anisotropic model yielded much better accuracy when compared to a number of phenomenologically-based constitutive laws, as demonstrated by its significantly reduced least-square errors in stress curve fitting. The results confirmed that the human GB wall material was heterogeneous, particularly toward the apex region. Our study also suggested that non-uniform wall thickness of the GB was important in determining the material parameters, in particular, on the parameters associated with the properties of the matrix and the longitudinal fibers-the difference could be as large as 20-30% compared to that of the uniform thickness model.

3D physiological model of the aortic valve incorporating small coronary arteries.

The diseases of the coronary arteries and the aortic root are still the leading causes of mortality and morbidity worldwide. In this study, a 3D global fluid-structure interaction of the aortic root with inclusion of anatomically inspired small coronary arteries using the finite element method is presented. This innovative model allows to study the impact and interaction of root biomechanics on coronary hemodynamics and brings a new understanding to small coronary vessels hemodynamics. For the first time, the velocity profiles and shear stresses are reported in distal coronary arteries as a result of the aortic flow conditions in a global fluid-structure interaction model.